Dispersion Models for
Geometric Sums
Bent Jørgensen ⋅ University of Southern Denmark
Célestin C. Kokonendji ⋅ Université de Franche-Comté, France
SDU, November 2010
Theme: Constructing geometric dispersion models
Convolution
Geometric compounding
Natural exponential families
Geometric tilting families
Exponential dispersion models Geometric dispersion models
Tweedie dispersion models
Geometric Tweedie models
Tweedie convergence
Geometric Tweedie convergence
See Jørgensen (1997) for background on dispersion models etc.
Geometric compounding
● Let X 1 , X 2 , … be i.i.d. variables, independent of
the geometric variable Nq, q ∈ 0, 1, with
PrNq = k = q1 − q k−1 for k = 1, 2, …
● Define the geometric sum Sq by
Sq = X 1 + ⋯ + X Nq .
● Average sample size (mean number of terms)
ENq = q −1
● Mean of Sq
ESq = q −1 EX 1 .
● See Kalashnikov (1997).
Convolution and geometric compounding
Convolution
Geometric compounding
MGF
Mt = Ee tY 
Mt = Ee tY  MGF
CGF
κ = log M
c = 1 − e −κ
GCF
κ X 1 +⋯+X n
nκ X 1
q −1 c X 1
c X 1 +⋯+X Nq
Mean
EY = κ̇ 0
EY = ċ0
Mean
Variance VarY = κ̈ 0 GY = c̈ 0
Geovar
MGF = moment generating function
CGF = cumulant generating function
GCF = geometric cumulant function
Geometric cumulants
● Cumulant generating function (CGF) for X
κs = log Ee sX 
● Geometric cumulant function (GCF) for X
c X s = 1 − e −κs
● First geo-cumulant is the mean
ċ X 0 = κ̇ X 0 = EX
● Second geo-cumulant is the geovar
2
GX = c̈ X 0 = κ̈ X 0 − κ̇ 2X 0 = VarX − E X
● The geovar
GX = VarX − E 2 X
satisfies the inequalities
− E 2 X ≤ GX ≤ VarX
and the scaling property
GaX = a 2 GX for a ∈ R.
● Exponential distribution Expμ defined by GCF
cs = sμ or Ms = 1 − sμ −1 for sμ < 1
positive/degenerate/negative exponential variable
for μ > 0, μ = 0, μ < 0.
● X ∼ Expμ has mean μ and geovar GX = 0.
● GX = VarX − E 2 X measures deviation from exponentiality.
● Rényi’s Theorem (Law of Large Numbers)
d
qSq → Expμ as q ↓ 0.
● Note fixed point
X 1 ∼ Expμ  qSq ∼ Expμ.
The geometric distribution
● Geometric variable Nq has MGF
Ee sNq  = 1 − q −1 1 − e −s  −1 ,
corresponding to the GCF
c Nq s = q −1 1 − e −s 
● Mean and geovar
ENq = q −1 , GNq = −q −1 < 0.
● Geometric sum Sq has MGF
Ee sSq  = Ee Nqκs 
= 1 − q −1 1 − e −κs 
−1
= 1 − q −1 cs −1
● So the GCF of Sq is proportional to c,
c Sq s = q −1 cs.
● The first two geo-cumulants for Sq are multiplied by q −1
ESq = q −1 EX 1 ,
GSq = q −1 GX 1 .
Convolution
Geometric compounding
Natural exponential families
Geometric tilting families
Exponential dispersion models Geometric dispersion models
Tweedie dispersion models
Geometric Tweedie models
Tweedie convergence
Geometric Tweedie convergence
● Recall exponential tilting of CGF κ with domain Θ = domκ
κ θ s = κs + θ − κθ for s ∈ Θ − θ
● Geometric tilting of GCF c with domain ⊆ domc
c θ s = cs + θ − cθ for s ∈ domκ − θ.
Geometric tilting families
● Geometric tilting family GEμ has GCF c θ s = cs + θ − cθ
with mean and geovar
μ = ċ θ 0 = ċθ
GX = c̈ θ 0 = c̈ θ
● Geometric tilting of Laplace distribution has density function
1 xμ − |x | 2γ + μ 2
1
fx; μ, γ =
exp γ
for x ∈ R,
2
2γ + μ
GCF
γ 2
c θ s = s + μs
2
mean and geovar
EX = μ ∈ R,
GX = γ > 0
The v-function
● Recall the variance function for an NEF,
Vμ = κ̈ ∘ κ̇ −1 μ for μ ∈ κ̇ Θ
● Define the v-function for geometric tilting family GEμ by
vμ = c̈ ∘ ċ −1 μ for μ ∈ Ψ = ċdomain
(requires ċ to be monotone).
● v, Ψ characterize GEμ among all geometric tilting families:
c̈ s + θ
↗
↘
v = c̈ ∘ ċ −1
cs + θ − cθ  ċs + θ
↖
↙
ċ −1 + const. = ∫ 1/v
Quadratic v-functions
Family
vμ
Ψ
cs
Laplace
1
R
s 2 /2
Geometric
−μ
1, ∞
1 − e −s
Geometric Poisson
μ
R+
es − 1
Geometric gamma
μ2
R+
− log1 − s
R+
− log1 − e s 
R
− log cos s
Geometric negative binomial μ1 + μ
Geometric GHS
1 + μ2
GHS = Generalized Hyperbolic Secant distribution.
Convolution
Geometric compounding
Natural exponential families
Geometric tilting families
Exponential dispersion models Geometric dispersion models
Tweedie dispersion models
Geometric Tweedie models
Tweedie convergence
Geometric Tweedie convergence
● Geometric dispersion model GD ∗ μ, γ and GDμ, γ have GCF
s  γ −1 c θ s = γ −1 cs + θ − cθ (additive case)
s  γ −1 c θ γs = γ −1 cγs + θ − cθ (reproductive case).
Geometric dispersion models
∗
GD μ, γ GDμ, γ
GCF
γ −1 c θ s
γ −1 c θ γs
Mean
γ −1 μ
μ
γ −1 vμ
γvμ
Geovar
● Geometric reproductivity: For i.i.d. X 1 , X 2 , … ∼ GD ∗ μ, γ
Sq ∼ GD ∗ μ, qγ.
● Hence the dispersion parameter γ absorbs probability parameter q.
Geometric infinite divisibility
● A variable X is called geometric infinitely divisible if
for each q ∈ 0, 1 there exists a geometric sum Sq such that
d
X = Sq
● GD ∗ μ, γ and GDμ, γ are geometric infinitely divisible
if and only if γ has domain R + .
Exponential mixtures
● X is infinitely divisible if and only if it is an exponential mixture:
Ee sX |Z = e Zκs ,
where Z ∼ Expλ with λ > 0. Then X has MGF
Ee sX  = Ee Zκs  = 1 − λκs −1
and GCF λκs.
● If the CGF κ is infinitely divisible with variance function V, then
c = κ is a geometric infinitely divisible GCF with v-function v = V.
Convolution
Geometric compounding
Natural exponential families
Geometric tilting families
Exponential dispersion models Geometric dispersion models
Tweedie dispersion models
Geometric Tweedie models
Tweedie convergence
Geometric Tweedie convergence
● Tweedie ED model Tw p μ, γ has mean μ and unit variance function
for μ ∈ Ω p ,
where p ∉ 0, 1.
Vμ = μ p
● Recall scale transformation property
b −1 Tw p bμ, b 2−p γ = Tw p μ, γ
for
b > 0.
Geometric Tweedie models
● Geometric Tweedie model GT p μ, γ has mean μ and unit v-function
for μ ∈ Ω p
and p ∉ 0, 1.
vμ = μ p
● GT p μ, γ is an exponential mixture of Tw p μ, γ.
● Characterized by scale transformation property
b −1 GT p bμ, b 2−p γ = GT p μ, γ
for
b > 0.
Summary of Geometric Tweedie models
vμ = μ p
and
α = 1 + 1 − p −1
GT p μ, γ
p
α
Support
Geometric extreme stable models
p<0
1<α<2
R
Geometric Laplace models
p=0
α=2
R
Geometric Poisson models
p=1
α = −∞
N0
α<0
R0
Geometric compound Poisson models 1 < p < 2
Geometric gamma models
p=2
α=0
R+
Geometric Mittag-Leffler models
p>2
0<α<1
R+
Models with exponential v-functions
p=∞
α=1
R
Geometric Laplace model
● GT 0 μ, γ has vμ = 1 and density function
1 xμ − |x | 2γ + μ 2
1
fx; μ, γ =
exp γ
2γ + μ 2
for x ∈ R.
● Exponential mixture of normal distributions.
● Laplace convergence (like Central Limit Theorem)
−1/2
d
γ GDγ μ, γ → GT 0 μ, v0 as γ ↓ 0,
which follows from vμ ∼ v0 as μ ↓ 0 (see next section).
1/2
Geometric Mittag-Leffler model
● Define the CGF κ α for α ≠ 0, 1 by
α
α
−
1
s
α
for s/α − 1 > 0,
κ s = α
α−1
which is an extreme α-stable distribution. Note that
α
α
κ α
θ s = κ θ1 + s/θ − 1.
● Let Y = X 1/α Z, with X unit exponential and Z extreme α-stable:
M Y s = EexpsX 1/α Z = Eexpγ −1 κ α sX = 1 − γ −1 κ α s
see Kozubowski (2000).
● This generates the geometric Tweedie distribution.
● p > 2 gives the Mittag-Leffler distribution on R + .
● p < 0 gives the geometric extreme α-stable distribution on R.
−1
Geometric Poisson model
● Poisson CGF is κs = e s − 1,
is exponential mixture of Poissons with GCF
cs = e s − 1
and MGF corresponding to c θ s is
Ms = 1 − e θ e s − 1 −1 .
● This defines GT ∗1 μ, 1, a shifted geometric distribution.
● Note: θ is mixed up with γ.
● v-function vμ = μ for μ > 0.
Geometric compound Poisson model
● GT p μ, γ for p ∈ 1, 2 is an exponential mixture of compound
Poisson models.
● Special case GT 3/2 μ, γ has distribution function
Fx; μ, ρ = 1 − ρe −ρx/μ for x > 0,
where
1−ρ
γ=2 ρ
● This is a zero-modified exponential distribution with GCF
s
cs =
1 − s/2
Geometric gamma model
● Exponential mixture of gamma distributions
∞
1 x u−1 e −u du for x > 0,
−x
fx = e ∫
0 Γu
which has GCF
ct = − log1 − s.
This generates the geometric Tweedie model GT 2 μ, γ.
● Scaling property
b −1 GT 2 bμ, γ = GT 2 μ, γ for b > 0.
Note invariance of γ under scaling.
Exponential v-functions
● Exponential v-function
vμ = e βμ
Correspond to the power p = ∞ or α = 1.
● Exponential mixture of Tweedie models with exponential variance
functions.
Convolution
Geometric compounding
Natural exponential families
Geometric tilting families
Exponential dispersion models Geometric dispersion models
Tweedie dispersion models
Geometric Tweedie models
Tweedie convergence
Geometric Tweedie convergence
● If EDμ, γ has Vμ ∼ μ p as μ → 0 or ∞ then
−1
b EDbμ, b
2−p
d
γ → Tw p μ, γ
as b → 0 or ∞,
see Jørgensen, Martínez and Tsao (1994) and Jørgensen (1997).
● Proof: Use b −p vbμ → μ p and Mora’s convergence theorem.
Convergence of v-functions
● Given sequence GE n μ with v-functions v n . If
v n μ → vμ as n → ∞
uniformly on compact sets then
GE n μ → GEμ as n → ∞,
where GEμ has v-function v.
● If v ≡ 0 then the limiting distribution is Expμ.
● Proof: Follow backward arrows in diagram:
c n s + θ − c n θ  ċ n s + θ
vn
↖
↙
ċ −1
n + const. = ∫ 1/v n
Law of large numbers
● Law of large numbers for ED models:
p
EDμ, γ → μ as γ ↓ 0.
● Law of large numbers for GD models:
d
GDμ, γ → Expμ as γ ↓ 0,
because γvμ → 0 as γ ↓ 0.
● Implies Rényi’s Theorem:
d
qSq → Expμ as q ↓ 0.
Tweedie convergence as large-sample result
● Tweedie convergence again
−1
b EDbμ, γb
2−p
d
 → Tw p μ, γ
as b → 0 or ∞.
p−2
● Recall Ȳ n ∼ EDμ, γ/n, so take n = b n (p ≠ 2)
● n large implies b n large (p > 2) or small (p < 2).
● Hence consider large-sample Tweedie convergence
n
−1/p−2
EDn
1/p−2
d
μ, γ/n → Tw p μ, γ
as n → ∞.
● Scaled and tilted Ȳ n converges to Tweedie distribution.
Geometric Tweedie convergence
● If GDμ, γ has vμ ∼ μ p as μ → 0 or ∞ then
−1/2−p
d
GDq
μ, γq  GT p μ, γ
as q ↓ 0 or → ∞.
q
Scaled and tilted qSq ∼ GDμ, γq goes to geometric Tweedie
model.
1/2−p
● Proof: Use vμ ∼ μ p
q −2/2−p γqvn 1/2−p μ → γμ p
and apply convergence theorem for v-functions.
Summary
Convolution
Geometric compounding
Natural exponential families
Geometric tilting families
Exponential dispersion models Geometric dispersion models
Tweedie dispersion models
Geometric Tweedie models
Tweedie convergence
Geometric Tweedie convergence
Other forms of dispersion models
● Additive characterization of convolution ∗ via cumulant transform
κ X ∗n = nκ X
Convolution
Minimum
Free convolution
Ordinary
Geometric sum
Tweedie
Geometric Tweedie
Extreme value
?
?
?
● Use Vμ = κ̈ ∘ κ̇ −1 μ for characterization and convergence.
● Normal distribution: Rayleigh, Laplace distributions.
● Free exponential families (Bryc, 2008).
Normal distribution: Wigner’s semicircle law.
References
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